IMMIGRATION AND THE STABLE POPULATION MODEL

IMMIGRATION AND THE STABLE POPULATION MODEL Thomas J. Espenshade The Urban Institute, 2100 M Street, N. W., Washington, D.C. 20037, USA Leon F. Bouvier Population Reference Bureau, 1337 Connecticut Avenue, N. W., Washington, D.C. 20036, USA W. Brian Arthur International Institute for Applied Systems Analysis, A-2361 Laxenburg, Austria RR-82-29 August 1982 Reprinted from Demography, volume 19(1) (1982) INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS Laxenburg, Austria

Research Reports, which record research conducted at IIASA, are independently reviewed before publication. However, the views and opinions they express are not necessarily those of the Institute or the National Member Organizations that support it. Reprinted with permission from Demography 19(1): 125-133, 1982. Copyright 1982 Population Association of America. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage or retrieval system, without permission in writing from the copyright holder.

iii FOREWORD For some years, IIASA has had a keen interest in problems of population dynamics and migration policy. In this paper, reprinted from Demography, Thomas Espenshade, Leon Bouvier, and Brian Arthur extend the traditional methods of stable population theory to populations with below-replacement fertility and a constant annual quota of in-migrants. They show that such a situation results in a stationary population and examine how its size and ethnic structure depend on both the fertility level and the migration quota.

DEMOGRAPHY Volume 19, Number 1 Februory 1982 IMMIGRATION AND THE STABLE POPULATION MODEL Thomos J. Espenshade The Urban Institute, 2100 M Street, N.W., Washington, D.C. 20037 Leon F. Bouvier Population Reference Bureau, 1337 Connecticut Avenue, N.W., Washington, D.C. 20036 W. Brian Arthur International Institute for Applied Systems Analysis, A-2361 Laxenburg, Austria Abstract-This paper reports ()n work aimed at extending stable population theory to include immigration. Its central finding is that, as long as fertility is below replacement, a constant number and age distribution of immigrants (with fixed fertility and mortality schedules) lead to a stationary population. Neither the level of the net reproduction rate nor the size of the annual immigration affects this conclusion; a stationary population eventually emerges. How this stationary population is created is studied, as is the generational distribution of the constant annual stream of births and of the total population. It is also shown that immigrants and their early descendants may have fertility well above replacement (as long as later generations adopt and maintain fertility below replacement), and the outcome will still be a long-run stationary population. Since the beginning of the twentieth century, the population of the United States has roughly tripled-from approximately 75 million in 1900 to about 225 million in 1980. Both natural increase (births minus deaths) and net immigration (immigrants minus emigrants) have contributed to this growth. During the decade 1901-1910 the average annual number of immigrants to the United States was nearly 880,000, and net immigration accounted for 40 percent of intercensal population growth. 1 But following 1910 the importance of net immigration relative to natural increase declined, reaching a minimum during the Depression decade, 1931-1940, when emigrants outnumbered immigrants. The 1965 amendments to the 1952 Immigration and Naturalization Law replaced the previous annual ceiling of 154,000 immigrants with a preference system permitting 290,000 immigrants plus about 100,000 relatives of citizens to enter the country each year. The effect of these regulations was to increase substantially the volume of immigration, and for the next decade the annual number of legal immigrants was close to 400,000. Recent statistics indicate a further increase to perhaps 600,000 per year, including refugees. With this growth in numbers, the relative contribution of net immigration to overall U.S. population growth has once again risen; for the period 1971-1978, it was estimated at 22 percent. Falling birth rates have accentuated the rising comparative importance of net immigration. The U.S. total fertility rate crossed below the replacement level in 1972, for the first time since the Depression, and it has fluctuated around 1.8 or 1.9 ever since. Annual births still exceed annual deaths, but that is due to a temporary phenomenon of large proportions of females in the childbearing ages. We may ask what the U.S. population would look like if current conditions 125

126 DEMOGRAPHY, volume 19, number 1, February 1982 were to persist into the indefinite future. Specifically, suppose fertility and mortality schedules were held constant so that fertility was permanently below replacement, and suppose that a constant number of persons (with a fixed age distribution) migrate to the United States each year. Would the population continue to grow because of the influx of immigrants and the children they would bear? Would the population eventually level off and then experience a long-term decline owing to subreplacement f ertility? Or, would net immigration counterbalance the low fertility rates, causing a stationary population to evolve? This problem takes on added significance since immigration has been and is likely to continue to be an important source of U.S. population growth, and because immigration will be a major policy consideration throughout the 1980s. Moreover, the circumstance of below-replacement fertility plus net immigration is one shared by numerous other industrial nations. There are two ways to answer the question. One is with a straightforward projection of the U.S population. To illustrate this approach, we use the estimated U.S. population on July 1, 1977 and project it forward on the assumption that 1977 age-specific fertility and mortality rates remain constant and that net immigration totals 400,000 each year. Given these postulates, we arrive ultimately at a stationary population. As seen in Table 1, the eventual stationary population contains 107,903,100 persons, with 1,209,800 annual births and 1,609,800 annual deaths to offset the 400,000 immigrants. A second approach is to analyze the problem in terms of stable population theory. Typically, by assuming a female population closed to the influence of migration, the stable model has investigated the shape of the long-run age distribution and eventual levels for rates of birth, death, and natural increase when underlying age-specific fertility and mortality schedules are fixed. Here we add the assumption of a fixed annual number and age composition of immigrants. 2 Focusing on females, we may extend the theory to include immigration in the following way. STABLE THEORY WITH BELOW REPLACEMENT FERTILITY AND CONST ANT IMMIGRA TION 3 Annual Births If we represent the annual number of females immigrating at age a by I(a), the annual rate of bearing daughters for women at age a by m(a), and the probability of surviving from birth to exact age a in the female life table by p(a), then the annual number of births at time t, B(t), can be expressed as the sum across all ages of childbearing of the number of women at age a at time t multiplied by the annual rate of childbearing at age a, or as B(t) = J: N(a, t) m(a)da (I) where a and f3 denote the lower and upper limits of the childbearing ages, respectively. Since we are interested in the long-run character of the population, we will restrict our attention to values of t > {3, where t = 0 represents the time after which /(a), m(a), and p(a) are held constant. For t > {3, women in the population at time t = 0 are no longer bearing children, and the youngest females in the first wave of immigrants after t = 0 have reached the end of their childbearing years. The number of women at age a at time t depends first on the number of women who were born in the population a years earlier and have survived to age a, and second on the number of women who immigrated at all ages less than a and are now age a. The first component can be written as B(t - a) p(a). To understand the second component, consider a particular age, say age 23. Then the number of foreign-born women who are now age

Table 1.-U. S. Population, July I, 1977, and Eventual Stationary Population Achieved with Constant 1977 Fertility and Mortality and 400,000 Annual Immigrants (all numbers in thousands) Age U.S. Population, July 1, 1977 Immigration Assumptions Eventual Stationary Population Females 0-4 7414.0 5-9 8400.3 10-14 9413. 0 15-19 10428.0 20-24 9978.0 25-29 8909.0 30-34 7776.0 35-39 6309.0 40-44 5735.0 45-49 5898.0 50-54 6167.0 55-59 5766.0 60-64 4983.0 65-69 4708.0 70-74 3543.0 75+ 5634.0 Males 7760.0 8759.0 9791.0 10753.0 10111.0 8837.0 7640.0 6030.0 5465.0 5613.0 5714.0 5277.0 4380.0 3732.0 2594.0 3219. 0 Females Males 16.4 17.6 16.4 17.6 10.4 11.2 6.4 5.6 20.4 8.4 48.4 47.2 34.4 44.0 19.2 26.4 10.0 14.0 5.6 8.0 2.8 4.0 1. 6 2.0 0.4 0.8 0.2 0.3 0.1 0.1 0.1 0.1 Females Males 2952. 2 3089. 8 3027.3 3168.0 3090.6 3234.1 3126.3 3261.1 3183.5 3266. 6 3344.4 3372. 3 3537.7 3567.5 3651. 9 3703.2 3692. 8 3744.6 3680.2 3706.2 3622.3 3590.6 3517.1 3387.7 3352. 6 3078.6 3121. 4 2660.9 2798.5 2142.2 5994. 5 3236.4 i 3 ci5" a c; ::0 Q ::0 0.. :;. if CT ii".,, ~ c c; a ::0 :it &.!.. Total 111,061. 2 105,675.0 192.8 207.2 55,693.2 52,209.9 Both sexes 216,736.2 400.0 107,903.1 Swmnary Demographic Measures 1977-1982 Stationary Population Total fertility rate Gross reproduction rate Net reproduction rate (NRR) Male births per 100 female births Female life expectancy at birth (in years) Male life expectancy at birth (in years) 1. 826 0.891 0. 869 105.0 77.09 69.32 1.826 0.891 0. 869 105.0 77.09 69.32 Population size Yearly births Yearly deaths Yearly net immigrants 221, 241. 8 3,449.6 2,028.7 400.0 107,903.1 1,209.8 1,609.8 400.0 Annual rates per 1,000 population Birth rate Death rate Natural increase Net migration Population increase 15.6 9.2 6.4 1. 8 8. 2 11. 2 14.9-3. 7 3.7 o.o...!::i

128 DEMOGRAPHY, volume 19, number 1, February 1982 23 equals the number of females who migrated at age 0 times the probability of surviving from age 0 to age 23, plus the number of females who migrated at age l times the probability of surviving from age l to age 23, and so on. Expressing this algebraically, the number of foreignborn women who have attained age a at time t equals /(O). p(a) + /(l). p(a) p(o) p(i) p(a) + + /(a - 1) + /(a). p(a - 1) The continuous-form analog of this number is Therefore, f N(a, t) = B(t - a I(x) p(a) dx. o p(x) a) p(a) + fa I(x) p(a) dx. (2) o p(x) In words, equation (2) says that the number of women in the population who are age a at time t is the number of native-born women who have attained age a plus the number of foreign-born women who have attained age a. Since the second term on the righthand side of (2) depends only on a and not on t, it is simpler to write it as Hl._a). Now we can substitute for N(a, t) in (1) to obtain B(t) = J: B(t - a)p(a)m(a)da + J: Hl._a)m(a)da. (3) This equation tells us that the total number of births at time t is the sum of births to native-born women and births to foreign-born women. Since the second term on the right-hand side of equation (3) does not involve the variable time t, the number of births to foreign-born women is some constant value that is repeated year after year. We can represent it by Bi so that B(t) = J: B(t - a)p(a)m(a)da +Bi. (4) We may now ask what the long-run behavior of B(t) will be. Taking Laplace transforms across (4) in the usual way, we have - - - Bi B(s) = B(s) F(s) + - s where F(s) is given by F(s) = {'" e-sa p(a)m(a)da. From (4a) we obtain - Bi (4a) (4b) B(s) = _ (5) s(l - F(s)) We now invoke the tauberian theorem that, providing sb(s) has no singular points for s > 0, then liml--+"" B(t) = lims--+o sb(s). This means in our case that as long as I - F(s) does not equal zero for any positive s, which from (4b) is guaranteed only if J 0 00p(a)m(a)da < 1, then the birth trajectory must reach an asymptotic limit given by B Jim B(t) = Jim '- 1--+oo s->o (1 - F(s)) l - f 00 0 p(a)m(a)da (6a) We recognize f 0 00p(a)m(a)da as the net rate of reproduction NRR. The theorem thus tells us that providing the NRR < 1,

Immigration and the Stable Population Model births must ultimately level off to a constant B given by B1 B=---- 1 - NRR (6b) The reader may check that a stationary level B does indeed satisfy (4) if that is, if B = L"" Bp(a)m(a)da + B1 as in (6b). To summarize, we have shown that the annual number of births eventually becomes stationary, at a level equal to the annual number of births to immigrant women divided by 1 - NRR. Total Population To calculate total population size, we return to equation (2) and recognize that the total number of females is obtained by adding up the number at each age, or that 129 represent the total size of the foreignborn population, f 0 whj(a)da. Thus, or (9) N = B1 ( eo ) + H1. (10) 1 - NRR Equation (9) shows that the total eventual stationary population is actually composed of two smaller constant populations. One of these arises from a constant annual number of births and has an exact parallel in the ordinary life table stationary population. There, the crude birth rate (loft 0 ) equals the reciprocal of life expectancy at birth (T o/1 0 ), so that the total population that would ultimately be generated by a constant yearly number of births (B) is B eo. The second population contains HJ. the stock of foreign-born women. We can compute H 1 simply, by summing HJ(a)-the number of immigrants in the population who are age a-across all ages. This yields: N(t) = Lw N(a, t)da where N(t) is the total number offt~µiales at time t, and w is the oldest age attained by anyone in the population. Substituting from (2) into (7) we have N(t) = f w {B p(a) + Ht(a)}da. (8) 0 Since the right-hand side of equation (8) does not involve the variable t, total population size does not change with time. We can therefore drop t from the left-hand side, knowing that we have a formula for the size of the eventual stationary population (N). It is possible to write equation (8) more simply by realizing that f o "'p(a)da is another way of expressing life expectancy at birth (e 0 ) and by letting H1 = I(x)-dxda. p(a) o o p(x) f wfa (11) Substituting for B 1 and H 1 in (10) we may write the total population size, in full, as N~ c- ~RR) fa f f3 p(a) I(x) - m(a)dxda a o p(x) w fa p(a) I + I(x)-- dxda. o o p(x) (12)

130 NUMERICAL RESULTS To confirm our analytic results, we have applied them to U.S. fertility and mortality schedules for 1977 and to the data in Table 1 on immigrants. The annual number of female births (B) in the stationary population is given by equation (6b) as Bi B=---- 1 - NRR where B 1, the annual female births to immigrants, can be evaluated using the second term on the right-hand side of equation (3). Doing so yields B 1 = 77.29 thousand, and combining this with NRR = 0.869, we have B = 77.29 -:-.131 = 590 thousand. In Table 1 annual male and female births combined total 1209.8 thousand, but since these projections assume a sex ratio at birth equal to 105 males per 100 females, approximately 0.4878 of all births are female. Therefore, the computer-based projections imply that B = 1209.8 x.4878 or 590. l thousand. Total female population size (N) is computed from equation (9) as N = Be 0 + H 1, where Hh the size of the foreignborn female population, is equal to fowhi(a)da. Setting B = 590.1, e 0 = 77.09, and H 1 = 10,201.25, we have N = 55,692. l thousand. This, except for rounding, is the same as the number in Table 1. For the female population the crude birth rate is 10.60, the crude death rate is 14.06, the immigration rate is 3.46, and the rate of natural increase equals -3.46. DISCUSSION AND FURTHER RESULTS If stable theory is expanded to include immigration, we have shown that as long as fertility is below replacement, a stationary population results by combining fixed fertility and mortality schedules with a constant number and age distribution for immigrants. Neither the level of the net reproduction rate nor the size of the annual immigration qualitatively af- DEMOGRAPHY, volume 19, number 1, February 1982 fects this conclusion; a stationary population eventually emerges. We can both generalize the above result and see how this stationary population is constructed, using a simple heuristic argument. Imagine a country divided into halves in such a way that the population alive at time t = 0 and any of its descendants reside in the western portion, and immigrant arrivals after t = 0 together with their descendants reside in the eastern portion. Concentrating first on the population in the west, we can see that this population eventually dies away. Even though it may continue growing for a while after t = 0 due to the momentum that a youthful age composition imparts to population growth, its below-replacement fertility is sufficient to guarantee a negative stable growth rate and, therefore, long-run extinction. The eastern portion of the country develops demographically in a more complex way. Any population that exists there must either be direct immigrants or the descendants of immigrants. Hence this population (that is, the female part of it) will consist at any time of surviving immigrant women, native-born women whose mothers were immigrants, nativeborn women whose grandmothers were immigrants, and so on. It will be useful to C8:~J women whose mother immigrated "first generation," whose grandmother immigrated "second generation," whose great-grandmother immigrated "third generation," and so on, tagging each woman in the population by her immigration ancestry. We can assume, in general, that fertility behavior differs for women of different immigration ''generations," so that women of "generation" i have fertility schedule m,{a), with associated net reproduction rate NRR;. The eastern population then builds up as follows. In a relatively short time after time zero, say two or three generations, the stock of surviving direct immigrants becomes constant and stays constant, building up in exactly the same way as a standard life-table population, except

Immigration and the Stable Population Model that in this case people can enter the population at all ages. In time, then, there is a constant number of surviving immigrant women H/.,_a) at age a, in any year. In tum, each year thereafter B 1 children are born whose mothers are immigrants, where B1 = J: H/.,_a)m0(a)da (13) and where m 0 (a) is the fertility schedule of immigrant women. Since immigrants are constant in number at any age, these annual "first generation" births are constant too. A generation or so after the appearance of "first-generation" births, "second-generation" births B 2 start to appear. Since these are born to the constant flow of "first generation" births, they number B2 = J: BJP(a)m1(a)da = NRR 1B 1 (14) and each year, they too are born in constant numbers. Given sufficient time, children of all "generations" up to "generation" Rare born each year, and generalizing (14), we can show that each year produces a constant flow Bi of "generation i" births, where Bi= NRR-1B;-1; 2 ::s i ::s R. (15) As we move indefinitely into the future, all "generations" are represented in the eastern population, and the annual birth flow can be written as the infinite sum of "generational" births B = B 1 + B2 + B3 + ( 16) or, substituting from (15) B = B 1 (I + NRR 1 + NRR1 NRR2 + NRR 1 NRR2 NRR 3 + ). (17) This series will converge providing that NRR; is less than one for all "generations," after some finite number n. In 131 other words, the birth flow in the eastern population eventually becomes stationary, providing only that immigrantdescended women adopt below-replacement fertility a finite number of generations after "arrival." Now each of these births, whatever its "generational" status, faces the same survival schedule, and so each birth flow Bi generates its own stationary population B,e 0 Counting the annual stock of surviving immigrants, H1' in with the "generational" population stocks, the eastern-half population levels off at the value N = eob10 + NRR1 + NRR, NRR2 + NRR 1 NRR2 NRR 3 + ) + H1. (18) We can conclude from this argument that stationarity can still come about even when immigrants and their close desendants have above-replacement fertility. All we require is that from some "generation" on, immigrant descendants adopt, like the native population, below-replacement fertility. If so, stationarity is guaranteed. 5 Returning to the special case of the previous sections, where all net reproduction rates are equal and below one, we see that (17) becomes B = B,(l + NRR + NRR 2 + NRR 3 +... ) (19) or B = B 1 (20) I - NRR' which is the same as (6b), so that (18) is a generalization of our previous result, (10). Equations (16}-(20) provide a basis for determining the ''generational'' distribution of total births and of total population. In the example in Table 1, there are 590.1 thousand female births each year in the stationary population. Since NRR = 0.869, the fraction I - NRR or 13. l

132 DEMOGRAPHY, volume 19, number 1, February 1982 percent are "first-generation" births; 11.4 percent (13.1 x.869) will be "second-generation" births, and so on. The total stationary population includes 55,693.2 thousand females, of which 10,201.3 thousand, or 18.3 percent, are immigrants. Since we have assumed that all females are subject to the same agespecific death rates, the size of the native-born population, Beo = (B 1 + B2 + + B; + )e 0, is distributed by generation in the same proportions as total births. Thus, 10.7 percent of all females are "first-generation," 9.3 percent are "second generation," and so forth. The distribution of total population by "generational" status is important because the preservation of native language, tradition, and culture is likely to be infuenced by whether one is an immigrant, the child of an immigrant, or the grandchild. Cultural heterogeneity will be more pronounced the lower is the value of NRR. This kind of analysis can also be of practical significance in helping to formulate immigration policy. The projection in Table l shows that 400,000 annual net immigrants lead eventually to a total population of 107.9 million, or 269.76 persons in the stationary population for every annual immigrant. Suppose the United States wanted to arrive at a stationary population as large as the 1980 population of approximately 226 million. Then, assuming 1977 fertility and mortality conditions and the age-sex composition of immigrants in Table 1, almost 840,000 annual net immigrants would be needed-a number that may not be far from the 1980 figure. (Of course, the population would increase to almost 300 million before falling to 226 million.) CONCLUSION In this paper we have shown that any fixed fertility and mortality schedules with an NRR below one, in combination with any constant annual number and age distribution of immigrants, will lead in the long run to a stationary population. The size and other characteristics of this eventual stationary population depend only upon our assumptions regarding fertility, mortality, and the agesex composition of immigrants, and are not influenced in any way by the population we begin with. Moreover, we have shown that this long-run stationary population is actually composed of many smaller stationary populations-one of immigrants themselves, one of "first-generation" descendants, and so on. The composition of the total stationary population by its socalled "generational status" can be computed from a knowledge of the specific fertility, mortality, and immigration assumptions. We have shown that these results can be obtained even when some "generations" have above-replacement fertility. All that is required to establish a stationary population in the long run is that, at some point in the generational chain of immigrant descendants, one generation and all those that succeed it adopt fertility below replacement. NOTES 1 These and subsequent statistics on the part played by immigration in U.S. population growth are contained in Bouvier (1981). 2 Since immigration is controlled in most countries, assuming that the number of immigrants is constant is preferable to assuming constant rates of immigration. 3 This development parallels earlier work by Ansley J. Coale (1972). Coale approached the problem by starting with a stationary population closed to migration and then inquired how much of a reduction in fertility would be required when immigration is added to maintain a stationary population with the same number of births. We begin at the other end, by assuming below-replacement fertility and show that, with immigration constant both in volume and in age composition, a stationary population evolves. Moreover, any belowreplacement fertility schedule, if held constant, leads to a stationary population when constant immigration is included. For an interesting application of Coale' s approach assuming a lower estimate of net immigration to the U.S., see Keely and Kraly (1978). 4 Where quotation marks are used, "generation" signifies a label on each woman marking her immigration ancestry. Without quotation marks, genera-

Immigration and the Stable Population Model tion signifies as usual either time elapsed or a particular population as measured reproductively from some initial event or population. 5 Valuable information on this important subject has been provided by Bean et al. (1980). In their analysis of 1970 census data for Mexican-Americans, women are distinguished according to whether they were born in Mexico (first-generation), whether they were born in the United States but one or both of their parents were born in Mexico (second-generation), or whether they and their parents were born in the United States (third or higher generation). For ever-married Mexican American women aged 20-34, the average number of own children under age 3 (a measure of current fertility) was 0.64 for first-generation women, 0.57 for second-generation, and 0.53 for third or higher generation. By comparison, the average was 0.45 for non-mexican-american whites. When such other factors as age, education, and family income were controlled, first- and second-generation Mexican-Americans exhibited current fertility that was approximately 16 and 12 percent greater, respectively, than that of non-mexican-american whites. But for third or higher generations, the differences between Mexican-American fertility and that of other whites was not statistically significant. The authors conclude, " Hence, with respect to current fertility, later generational Mexican-American women, ceteris paribus, do not appear to behave differently from other white women" (p. 37). ACKNOWLEDGMENTS The authors thank Rachel Eisenberg Braun and an anonymous referee for 133 their helpful comments. The capable technical services of Bobbie Mathis are also appreciated. Views or opinions expressed in this paper are the authors' and do not necessarily represent those of the organizations with which they are affiliated. REFERENCES Bean, Frank D., G. Swicegood, and T. F. Linsley. 1980. Patterns of Fertility Variation Among Mexican Immigrants to the United States. Paper prepared for the Select Commission on Immigration and Refugee Policy, Washington, D.C. Texas Population Research Center Paper No. 2.016, The University of Texas at Austin. Bou vier, Leon F. 1981. The Impact of Immigration on U.S. Population Size. Population Trends and Public Policy, No. I. Washington, D.C.: Population Reference Bureau, Inc. Coale, Ansley J. 1972. Alternative Paths to a Stationary Population. Pp. 589-603 in Charles F. Westoff and R. Parke, Jr. (eds.), Demographic and Social Aspects of Population Growth, Research Reports, Volume I, U.S. Commission on Population Growth and the American Future. Washington, D.C.: U.S. Government Printing Office. Keely, C. B., and E. P. Kraly. 1978. Recent Net Alien Immigration to the United States: Its Impact on Population Growth and Native Fertility. Demography 15:267-283.